The correspondence principle states that the quantum system will approach to the classical system in high quantum numbers. Indeed, the average of the probability density distribution reflects a classical-like distribution. However, the likelihood of finding a particle at node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories which are obtained by solving the stochastic differential equation under the optimal guidance law. It turns out that point set A which is collected by the intersections of complex random trajectories and the real axis can present the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a distribution without appearance of nodes. Moreover, point set B can represent the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker-Planck equation.

中文翻译：

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对应原理的轨迹解释：节点问题的解决

对应原理指出，量子系统将接近高量子数的经典系统。实际上，概率密度分布的平均值反映了类似经典的分布。然而，在波函数的节点找到粒子的可能性为零。这种情况被认为是节点问题。在本文中，我们通过求解最优导律下的随机微分方程得到复量子随机轨迹来解决这个问题。结果表明，复随机轨迹与实轴的交点所收集的点集A可以呈现量子谐振子系统的量子力学相容分布。同时，复量子随机轨迹在实轴上的投影形成点集 B，该点集给出了没有节点出现的分布。此外，点集 B 可以表示高量子数中的经典相容分布。此外，通过Fokker-Planck方程的解验证了点集B的统计分布。